I think that you're missing the point.
I have a family line that goes back literally for ever. My father had a father. Obviously. And his father had a father. And we can take that line back as far as you want. So we can deduce that there is a direct paternal line between someone who lived a million years ago and me. Who that was, where he lived, what he looked like...we have no idea, although we could make some educated guesses.
Now there are, almost certainly (but not definitely) some people alive today who share a direct paternal line from that ancestor. So he would be the last common ancestor.
Yes, I get the argument. It's all fine until the ancestral line passes out of our realm of knowledge. From there it's speculation. Within engineering, good practice is not to extrapolate. Extrapolation is, at best, a hypothesis. If the hypothesis is promising, then you are tasked with getting the necessary data to demonstrate it. If the necessary data is below the noise floor ... too bad. Move on. My understanding is that such practice is common to all the sciences as well.
Why does there have to be a beginning? Maybe the ancestral line is infinite. I wouldn't argue such a thing, but it is an alternative. It's also OK to say, "We don't know."
Now if we propose that life only arose only started once ...
Exactly. If biologists want to make LUCA a postulate, fine. The problem seems to be, though, that such would allow others to propose other postulates (as in Euclidean vs Hyperbolic geometry). That makes biology untidy. We have all these branches on the tree that don't connect ... except beyond the noise floor where LUCA lives. It's like leaving your left flank unprotected ... and with all those religious nutters out there, well, we just can't have that.
I'm absolutely convinced biologists could come up with a scientifically grounded alternative to LUCA that would not require invoking God. But what motivation is there to do that?
Again, as I understand it, biologists don't think life started only once. Another conversation of mine from the past:
Biologist (paraphrasing): If the conditions were right at time X for life to start, I would be shocked if the result wasn't life starting thousands of times during time period X.
J_B_: Cool! (hope rising)
Biologist: But of course, only one of those thousands of biogenesis events survived.
J_B_: Oh. (deflating)
Biologist: Obviously.
J_B_: Uh huh. Obviously.